The asymptotic expansion of the regular discretization error of Itô integrals

Abstract

We study an Edgeworth‐type refinement of the central limit theorem for the discretization error of Itô integrals. Toward this end, we introduce a new approach, based on the anticipating Itô formula. This alternative technique allows us to compute explicitly the terms of the corresponding expansion formula. Two applications to finance are given; the asymptotics of discrete hedging error under the Black–Scholes model and the difference between continuously and discretely monitored variance swap payoffs under stochastic volatility models.

Asymptotic error distribution for the Riemann approximation of integrals driven by fractional Brownian motion

“…Proof of Proposition 3.5. We only do the proof of point (2), since the proof of point (1) (which requires to show that (u, P ) with P s := D s u s verifies the assumptions of Theorem 1.2) is very similar and easier. Before going into the details, let us explain the main steps we are going to follow:…”

“…The framework of Theorem 1.2 is general (assuming that the pair (u, P ) belongs to C 1 and satisfies other technical conditions) and concerns the convergence of M n,i,j as n → ∞ in probability, towards an identified limit. The situation where H > 1 2 differs significantly from H = 1 2 , because in this latter case M n,i,j converges in law (but not in probability, because of the creation of an independent alea, see e.g. (1.1)).…”

“…For example, we can mention multidimensional extensions (see [18]), generalizations to the case of random discretisation times (see [9]), applications in finance (see [11]) and approximation schemes of stochastic differential equations (SDEs) driven by semimartingales (see [14]). The recent paper [1] provides an asymptotic expansion for the weak discretization error of Itô's integrals.…”

“…. , B d ) of Hurst index H ∈ 1 2 , 1 (as already mentioned, all the processes considered in this paper are implicitly assumed to be measurable with respect to B);…”

“…Let us now study the fluctuations of M n,i,j • around its limit. 1), and let Z = (Z k,j ) 1≤k,j≤d (resp. W = (W k,j ) 1≤k,j≤d ) denote the matrix-valued Rosenblatt process measurable with respect to B (resp.…”